What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

It would be nice to have a strategy for disentangling any tangled ropes...

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

Can you find the area of a parallelogram defined by two vectors?

Can you describe this route to infinity? Where will the arrows take you next?

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

Can you find sets of sloping lines that enclose a square?

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 � 1 [1/3]. What other numbers have the sum equal to the product and can this be so. . . .

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Can all unit fractions be written as the sum of two unit fractions?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

Can you see how to build a harmonic triangle? Can you work out the next two rows?

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

What is the total number of squares that can be made on a 5 by 5 geoboard?

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand. . . .